December 12, 2013

Banker finds a duplication in a 3-manifold table

Daniel Moskovich recently wrote about the discovery by a lawyer of a duplication in the knot tables called the “Perko pair”.

Now a banker has found another duplicate in yet another table of 3-manifolds. This time it was Ben Burton, and the duplicate appears in the Hildebrand-Weeks cusped hyperbolic census.

I’m exaggerating a little. Ben is no longer a banker. But the duplication is real. The problem is that SnapPea’s Epstein-Penner decomposition code sometimes has trouble with highly symmetric manifolds. If the Epstein-Penner decomposition is not a triangulation, SnapPea attempts to build a subdivision. Unfortunately it does not always build a canonical-such subdivision. In this particular case, SnapPea generated distinct 5-tetrahedron subdivisions of a cube. Somehow the duplication was missed.

The problem resulted in a repeated manifold in the census.

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Thanks, but I don’t think I can be credited with that! It’s always been known that round-off error is a fact of life with certain parts of SnapPea, although (i) it hardly ever causes an issue, and (ii) I’m not sure that it caused the current problem, as tweaking a certain tolerance CONCAVITY_EPSILON in the SnapPea code doesn’t appear to change the output.

However, SnapPea does have “shake-down” combinatorial procedures randomize_triangulation(), basic_simplification(), and application of these did produce identical cell decompositions for the two manifolds. So SnapPea, used a little more intensively, does succeed in showing that these manifolds are homeomorphic.

Perhaps I could take the opportunity to emphasize, on a related issue, that my confidence level in the completeness of the extension of the cusped census to eight tetrahedra is distinctly less than 100%: there were many cases to consider, and simple clerical error could have led to omissions. I’d welcome an independent tabulation of these, as well as non-alternating knots of 17 to 20 crossings!

Hi Morwen: I’m in the process of writing up larger work on the snappea census now, and I can confirm that the 8-tetrahedron census is complete – my tabulation matches yours, and it uses exact computation to prove that nothing was omitted (i.e., no manifolds are missing because of numerical errors).

It appears that Thistlethwaite has achieved a new world record for knot tabulation. An independent jury has reliably confirmed that he (and Dowker) successfully extended an existing table, in this case through 12-crossings, without a single duplication [W.E.Clark, M.Elhamdadi, M.Saito and T.Yeatman, Quandle Colorings of Knots and Applications, arXiv:1312.3307v1 [math.GT] 11 Dec 2013]. The prior record holder was P.G.Tait, who did it through alternating 10’s in 1885.

Some history on the Knot Tabulation without Duplication competition (remarkable runners-up).

C.N.Little (Yale Ph.D.) was an indefatigable competitor, but hit the hurdles on three separate occasions: in 1885 (for alternating 10’s), 1890 (alternating 11’s) and yet again in 1899 (non-alternating 10’s). His first attempt did, however, enable Tait to walk away with the award, as Hoste. Thistlethwaite and Weeks report on page 35 of The Mathematical Intelligencer 20 (1998).

J.H.Conway, who did the work in high school, seems to have not looked closely at the 10-crossing tables, which had been cross-checked so long before by Little and Tait. Apparently the Great Student Revolution of the 1960’s came a bit late to England. Remember, kiddies: Always question authority!

Alain Caudron, whose pioneering work with arborescent knots allowed Bonahon and Siebenmann to classify them (in 351 pages!) probably shouldn’t count, since he only found two additional knots and his original link table, produced at the same time, contained a duplication. He was, however, a brilliant innovator. Too bad nobody could find him a decent job in mathematics.